Loop Quantization versus Fock Quantization of p-Form Electromagnetism on Static Spacetimes

TL;DR

This paper rigorously constructs Wilson loop quasioperators for vacuum electromagnetism on static spacetimes, advancing the mathematical understanding of loop quantization without regularization.

Contribution

It introduces a new framework for defining Wilson loops as quasioperators on Fock space in static spacetimes, addressing issues like noncompactness and Aharonov-Bohm modes.

Findings

Wilson loops are represented as sesquilinear forms, not operators.

The construction applies to p-form electromagnetism and Wilson surfaces.

Addresses technical issues in defining electromagnetic modes on noncompact spaces.

Abstract

As a warmup for studying dynamics and gravitons in loop quantum gravity, Varadajan showed that Wilson loops give operators on the Fock space for electromagnetism in Minkowski spacetime - but only after regularizing the loops by smearing them with a Gaussian. Unregularized Wilson loops are too singular to give densely defined operators. Here we present a rigorous treatment of unsmeared Wilson loops for vacuum electromagnetism on an arbitrary globally hyperbolic static spacetime. Our Wilson loops are not operators, but "quasioperators": sesquilinear forms on the dense subspace of Fock space spanned by coherent states corresponding to smooth classical solutions. To obtain this result we begin by carefully treating electromagnetism on globally hyperbolic static spacetimes, addressing various issues that are usually ignored, such as the definition of Aharonov-Bohm modes when space is noncompact. We then use a new construction of Fock space based on coherent states to define Wilson loop quasioperators. Our results also cover "Wilson surfaces" in p-form electromagnetism.